An Ab-initio Study - Horizon Research Publishing

Universal Journal of Mechanical Engineering 6(2): 21-37, 2018 DOI: 10.13189/ujme.2018.060201

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First Principles Study of the New Half-metallic Ferromagnetic Full-Heusler Alloys Co2CrSi1-xGex: An Ab-initio Study I. Asfour*, D. Rached Laboratory Magnetic Materials, Physics Department, Faculty of Science, Djillali Liabes University, Algeria

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Abstract We present an ab-initio study of the structural, electronic, elastic, magnetic, thermal and thermodynamic property of the quaternary Heusler alloys Co2CrSi1-xGex (x = 0, 0.25, 0.5, 0.75, 1) with the linearized augmented plane wave method based on density functional theory (DFT) and implemented in WIEN2k code. For exchange correlation potential we have used the generalized gradient approximation (GGA) of Perdew et al. Our results provide a theoretical study for the mixed Heusler Co2CrSi1-xGex (0 0, C44> 0, C11 + 2C12> 0 and C12< B < C11 [22]. The computed elastic constants satisfy the above stability criteria, indicating that these compounds are

Universal Journal of Mechanical Engineering 6(2): 21-37, 2018

elastically stable. The variation of the elastic constants works as a function of composition x. It is seen that, the elastic constants C11, C12, C44 decreases monotonically with x, the alloy structure becomes mechanically stable. The value of the Poisson’s ratio is indicative of the degree of directionality of the covalent bonds. Our obtained values for Poisson’s ratio vary from 0.369 to 0.372, which is an indication that the interatomic forces are central forces [23]. From our calculated values of the Zener anisotropy factor A, which is a measure of the degree of elastic anisotropy of the crystal, we note that our Heusler alloys are elastically isotropic The Young’s modulus E and Poisson’s ratio m are important in technological and engineering application [24]. Young’s modulus is defined as the ratio of stress and strain when Hooke’s law holds. The Young’s modulus of a material is the usual property used to characterize stiffness. The higher the value of E, the stiffer is the material. From Table 1, the Young’s modulus decreases when we move from 0 to 1. Thus, the Co2CrSi is stiffer than Co2CrGe. The Debye temperature is known to be an important fundamental parameter closely related to many physical properties, such as specific heat and melting temperature. At low temperatures, the vibrational excitations arise solely from acoustic vibrations. Hence, at low temperatures the Debye temperature calculated from elastic constants is the same as that determined from specific heat measurements. Table 1. Calculated lattice constant (a), bulk modulus (B), pressure derivative of the bulk modulus (B’), elastic constants (Cij), shear modulus (G), ratio of B/G, Young’s modulus (E), Poisson’s ratio (ν), zener anisotropy factor (A), density (q), longitudinal elastic wave velocities (Vl), transverse elastic wave velocities (Vt), average acoustic velocity (Vm), and Debye temperature (θD) of Co2CrSi1-xGex for various compositions a(Å) B (Gpa) B’ C11(GPa) C12(GPa) C44(GPa) G(Gpa) B/G E(GPa) ν A ρ(g/cm3) Vl(m/s) Vt(m/s) Vm(m/s) θD(K)

Co2CrSi0.75Ge0.25 Co2CrSi0.50Ge0.50 6.4021 6.4192 116.9321 114.8143 4.1243 4.3630 255.9906 246.3009 147.6590 139.3190 50.2373 47.2059 51.0087 48.9197 3.5045 3.4756 144.2141 138.3910 0.3697 0.3687 0.9643 0.9198 4.2892 4.5104 7543.6908 7187.8559 3375.1800 3223.3224 3907.0055 3739.2980 354.1048 334.0897

Co2CrSi0.25Ge0.75 6.4401 111.7985 4.1957 234.6867 135.5527 44.2890 45.6002 3.5839 129.6495 0.3723 0.9338 4.7209 6867.8643 3042.2204 3545.6822 312.4310

Fig 3 shows the variation of the lattice parameter calculated based on the concentration of germanium for the quaternary alloy. A slight deviation from the Vegard's law is clearly visible for the alloy with bowing upwardly parameter equal to 0.0024 Å, obtained by adjusting the values calculated by a polynomial function. The physical origin of this small gap could be mainly due to the low disparity Co2CrSi of lattice constants and ternary

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compound Co2CrGe. The equation that represents the variation of the lattice parameter depending on the concentration of Germanium is: A=6.2049+0.3910x+0.0024x2

Figure 3. Variation of the lattice parameter of Co2CrSi1-xGex Heusler alloys with Ge concentration.

Fig 4 represents the variation of the bulk modulus as a function of the germanium concentration in the Co2CrSi1-xGex alloy. A significant deviation of the bulk modulus is observed with a disorder parameter equal to 8.243 GPa for Co2CrSi1-xGex alloys, show that the incompressibility modulus decreases with increasing the concentration of Ge (0 ≤ x ≤ 1). The equation that represents the compressibility modulus of variation depending on the concentration of Germanium is: B=192.8820-40.6745x+8.2430x2

Figure 4. Variation of the Bulk modulus of Co2CrSi1-xGex Heusler alloys with Ge

3.3. Electronic Properties To elucidate the nature of the electronic band structure, we also calculated the total and partial densities of states, as shown in Figures 6 most transport properties are

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First Principles Study of the New Half-metallic Ferromagnetic Full-Heusler Alloys Co2CrSi1-xGex: An Ab-initio Study

determined on the basis knowledge of the density of states. Generally, in the vicinity of the Fermi level, the bands are mainly due to orbital states d of Cr and the transition metals; this is justified by the half metallic character. Densities of states confirming the metallic character for the majority of densities are presented in Figure 5. A lack of electronic states at the Fermi level of minority spins us closer to the semiconductor and semi-metallic character for both alloys. From Figure 6, it is clear that there are three distinct regions in the spin-up state and in the spin-down state separated by gaps materials. We notice an overlap valence and conduction bands to the state spin-up, while the existence of electronic states at the Fermi level tells us about the nature of these metal alloys. The presence of electronic states is most obvious for the spin up, it is minimal for the spin down and shows the near character semiconductor, and in this state we see a gap between the conduction and the valence band. It can be said that the compounds and their alloys are semiconductors in the state spin-dn. This means that the system has a half-metallic character. We notice a change in indirect gap for quaternary alloy materials studied.

Figure 5. Total and partial density of states of the quaternary Heusler alloys Co2CrSi1-xGex.


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Figure 6. Calculated band structures of the quaternary Heusler alloy Cr2GdSn1-xSbx

We notice an overlap valence and conduction bands to the state spin-up, while the existence of electronic states at the Fermi level tells us about the nature of these metal alloys. The presence of electronic states is most obvious for the spin up against by it is minimal for the spin down and shows the near character semiconductor. Note that the maximum of the valence band and the minimum of the conduction band are located at the same point X, so this is characteristic of direct gap. It can be said that the compounds and their alloys are semiconductors in the state spin-dn. This means that the system has a half-metallic character. Fig. 6 shows calculated band structures of the quaternary Heusler alloy Co2CrSi1-xGex The values of the gap energy are summarized in Table 2. Table 2. Calculated energy gap Co2CrSi1-xGex with concentration x of Ge. Compound

Co2CrSi1-xGex

x

Eg(eV)

0.00

0.8597

0.25

0.7949

0.50

0.6982

0.75

0.6547

1.00

0.646

3.4. Magnetic Properties The magnetic moment per formula unit for the alloys of 3d elements can be determined from the number of their valence electrons using Slater–Pauling (SP) rule [25].According to this rule, magnetic moment for the half metallic moment for the half metallic full-Heusler alloy is given by Mt=Zt-24, where Mt is the total magnetic moment per unit cell and Zt is the total number of valence electrons. The magnetic moment obtained by present calculations is

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4.0075 μB and 4.0020 μB for Co2CrSi and Co2CrGe respectively which agrees well with values from Slater– Pauling rule [25] as well as other calculations [26–28]. The energy band structure of a semi-metallic material has an asymmetry between the spin up states and spins down with a gap or a pseudo energy gap at the Fermi level. This gives rise to the polarizations of the conduction electrons at the Fermi level which may reach 100%. The calculated total and local magnetic moment for constituent elements are given in Table 3. Table 3. Calculated total and partial magnetic moments (in μB) Co2CrSi1-xGex Magnetic Moment (μB)

Compounds

Co Cr Si Ge Interstitial Moment Total (μB) Polarisation de spin %

Co2CrSi1-xGex X=0.25

X=0.50

X=0.75

1.0166 1.91019 -0.01562 -0.00256 0.07672 4.00152 100

0.9953 1.93610 -0.01984 -0.00899 0.8964 4.0019 100

0.99538 1.9549 -0.00630 -0.0284 0.08934 4.0003 100

It is evident that the total magnetic moment Increases with increasing pressure. The total magnetic moment is constant with increasing composition x).It is important to emphasize that, to our knowledge; the scientific community has no experimental or theoretical value of the magnetic moments in these materials.

Figure 7. The total magnetic moment as a function of the different lattice parameter for Co2CrSi1-xGex(x=0.25, 0.50, 0.75).

3.5. Thermal Properties We can calculate the thermodynamic quantities for the Heusler alloy Co2CrSi1-xGex through the Debye model

quasi-harmonic implemented in the Gibbs program [29] in which the non-equilibrium Gibbs function G* (V, P, T). Through the quasi-harmonic Debye model, one could compute the thermodynamic quantities of any temperatures and pressures of Co2CrSi1-xGexfrom the calculated E–V data at T = 0 and P = 0. (T being the temperature and P the pressure). The relationship between the lattice parameter and the temperature at different pressures is shown in Figure 8 the lattice parameter increases to a very moderate with temperature. On the other hand, it is noted in Figure 9 that the relationship between the bulk modulus and the pressure is virtually linear. The bulk modulus increases with pressure and decreases with temperature. Further, the temperature effects on the lattice parameters of the quaternary Heusler alloys Co2CrSi1-xGex with the Al concentration 0,0.25, 0.5, 0.75, 1 are shown in Figure. 13. As expected the volume increases with increasing temperature and the rate of increase is high. In Figure. 14 we report the evolution of the bulk modulus as a function of T at different concentrations. It is worth noting from the regular spacing of the curves observed in Figure.15 that the relationship between the bulk modulus and the temperature is nearly linear at various concentrations ranging from 0 to 1. The investigation on the heat capacity of crystals is an old topic of the condensed matter physics with which illustrious names are associated. Knowledge of the heat capacity of a substance not only provides essential insight into its vibrational properties but is also mandatory for many applications. Two famous limiting cases are correctly predicted by the standard elastic continuum theory [30]. At high temperatures, the constant-volume heat capacity Cv tends to the Petit and Dulong limit [31], the Figure 10 show the variation of the heat capacity Cv contant volume depending on the temperature for different pressures. This quantity indicates a strong increase up to ~ 500 K, which is due to anharmonic approximation Debye model. Values are purely predictive since we have no experimental data. However, at higher temperatures and at higher pressures, the effect of anharmonic Cv is deleted, and Cv tends towards the limit of Dulong-Pettit. (Cv (T) ≅ 74.60J.mol-1.K-1-74.46J.mol-1.K-1, respectively Co2CrSi and Co2CrGe is shown in figure 11. In figure 11 we present the variation of the Debye temperature θD as a function of temperature and pressure, respectively, one can observe that θD is nearly constant from 0 to 100K and decreases linearly with increasing temperature from T > 200 K. It can be seen that the Debye temperature θD increases with pressure and decreases with temperature.


Figure 8.

Variation of the lattice parameter as a function of temperature and pressure in Co2CrSi and Co2CrGe alloys.

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Figure 9.

Variation of the bulk modulus as a function of temperature and pressure in Co2CrSi and Co2CrGe alloys.


Figure 10.

Variation of the heat capacity as a function of temperature and pressure in Co2CrSi and Co2CrGe alloys.

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Figure 11. The Debye temperature θD as a function of temperature and pressure in Co2CrSi and Co2CrGe alloys.

The coefficient of thermal expansion α has been predicted figure.13 He quickly believes in cube power of temperature then tends towards a limit. We note, for a given pressure α increases with temperature when T≤300K (at low temperature), especially at zero pressure, and tends gradually to increase linearly at higher temperatures. As the pressure increases, the variation of α with the temperature becomes smaller. For a given temperature, α decrease sharply with increasing pressure, and is very low at higher temperatures and higher pressures as well.


Figure 12.

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Variation temperature coefficient as a function of temperature for different pressures for Co2CrSi and Co2CrGe

The vibrational properties that are related to the thermal effects are the heat capacity CV and the Debye temperature. Our results for the alloys concerning the heat capacity CV at different temperatures depicted in Figure 16. At high temperatures, the heat capacity CV approaches the classical value of 74.60 J.mol-1.K-1

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Figure 13.

Variation of the lattice parameter as a function of temperature and Al fraction in Co2CrSi1-xGex alloys

Figure 14. Variation of the bulk modulus as a function of temperature and Al fraction in Co2CrSi1-xGex alloys


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Figure 15. Variation of the heat capacity as a function temperature and Al fraction in Co2CrSi1-xGexalloys

3.6. Thermodynamic Properties Let us now turn our attention to the phase stability of Co2CrSi1-xGexquaternary Heusler alloys For this purpose, we calculate the Gibbs free energy of mixing ΔGm (x, T ), which allows accessing the T–x phase diagram and obtaining the critical temperature Tc. More details of calculations are given in [33–34] The Gibbs free energy of mixing, for an alloy is expressed as ΔGm=ΔHm-TΔSm Where ΔHm=Ωx(1-x) ΔSm=-R[xlnx+(1-x)ln(1-x)] ΔHm and ΔSm are the enthalpy and the entropy of mixing, respectively; Ω is the interaction parameter that depends on the material, R is the ideal gas constant and T is the absolute temperature. The mixing enthalpy of alloys can be obtained as the difference in energy between the alloy and the weighted sum of the constituents: ΔHm=E ABx C1-x –xEAB – (1-x) EAC Where E ABx C1-x, EAB and EAC are the energies of ABxC1-x, AB and AC materials, respectively. We calculate ΔHm to obtain the interaction parameter Ω as a function of the alloy concentration x. The variation of Ω as a function of the composition x for Co2CrSi1-xGex

quaternary Heusler alloy has been determined Figure. 16 The best fit of our data regarding Ω is found to be linear, yielding the following expression: Ω (kcal mol-1) =7.547-1.624x. The average value of the x-dependent Ω in the composition range 0 ≤ x ≤ 1 is estimated to be 6.737kcal*mol-1.Thelargerenthalpy of Co2CrSi1-xGex alloy suggests a large value of Ω and hence a higher critical temperature. Next, in order to determine the stable, metastable and unstable mixing regions of the alloy of interest, we have calculated the temperature–composition phase diagram. Our results are displayed in figure 17. At a temperature lowers than the critical temperature Tc, the two binodal points are determined as those points at which the common tangent line touches the ΔGm curves. Whereas the two spinodal points are determined as those points at which the second derivative of ΔGm is zero: δ2(ΔGm)/δx2= 0. A critical temperature (Tc) value of 981.274 K has been evaluated for Co2CrSi1-xGex.The equilibrium solubility limit, i.e. the miscibility gap, is marked by the spinodal curve in the phase diagram. For temperatures and compositions above this curve, a homogeneous alloy is predicted. One can also note the existence of a wide range between spinodal and binodal curves, thus indicating that Co2CrSi1-xGex may have a metastable phase. Hence our results indicate that the alloy Co2CrSi1-xGexis stable at high temperature.

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Figure 16.

Variation of the interaction parameter of Co2CrSi1-xGex Heusler alloys with Ge Concentration

Figure 17.

T-x phase diagram for Co2CrSi1-xGex alloys.

3.7. Enthalpy of Formation The enthalpy of formation of a crystal ΔHform is defined as the difference between the energy of the crystal and the sum of energies of the constituent elements that crystal in their standard states. To determine the thermodynamic stability and estimate the possibility of synthesizing these alloys, the formation enthalpy can be calculated using the following relationship: ΔHform(Co2CrZ)= Etotal (Co2CrZ) – [2Etotal(Co2) + Etotal(Cr) + Etotal(z)

Or Etotal (Co2CrZ) is the total energy of the compounds present in the L21 Phase. Etotal(Co2), Etotal(Cr) and Etotal(z) are the calculated total energies atoms in their standard states. Values training enthalpy for the investigated alloys are shown in Table 4. We can see that the enthalpy of formation take negative values, for all the compounds studied, calculated from the equation, means the existence and stability and it is possible to synthesize these alloys experimentally.


Table 4.

Values of enthalpy of formation ΔHform

Compound

Co2CrSi1-xGex

X 0.00 0.25 0.50 0.75 1.00

ΔHform(eV) -29.2287 -27.6563 -26.1622 -24.8058 -23.4974

4. Conclusions We have investigated in detail the structural, elastic, electronic, magnetic, thermodynamic and thermal properties of the quaternary Heusler alloys Co2CrSi1-xGex (x = 0, 0.25, 0.5, 0.75, 1) at ambient as well as at elevated temperatures. With the linearized augmented plane wave method based on density functional theory and implemented in WIEN2k code. For exchange correlation potential we have used the generalized gradient approximation (GGA) of Perdew et al. Our interest in this study was justified by the fact that the properties of these compounds are not available in the literature. The choice of compounds was warranted by the great deal of attention given to these Heusler alloys because of their large field of applications. Our calculated structural parameters are reasonable; also C11, C12 and C44 parameters were obtained from calculations. This quaternary Heusler alloy is mechanically stable according to the elastic stability criteria and shows ductile behavior. A linear variation of the lattice constant, elastic constants and Debye temperature with x has been obtained. The calculated lattice parameters for the alloys exhibit a tendency to Vegard’s law with a marginal bowing parameter. The electronic band structures show a half- metallic- character. The quasiharmonic Debye model is success-fully applied to determine the thermal properties at different temperatures and pressures. The results presented in this paper for the thermodynamic and thermal properties are predictions, and the experiments to prove them are welcomed. Finally, the calculated phase diagram indicated that Co2CrSi1-xGexis stable at temperature of 981.27 K.

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